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<p>We extend the notions of harmonic Bloch, harmonic Bloch-type mappings, and a linearly invariant family of harmonic mappings defined in the open unit disc 𝔻 of the complex plane to a hyperbolic domain of the complex plane. Then we establish the relation between linearly invariant family of holomorphic functions with harmonic Bloch mappings and a linearly invariant family of harmonic mappings with harmonic Bloch-type mappings defined on a hyperbolic domain of the complex plane. We also prove that the bounds of the Bloch seminorms for harmonic Bloch and harmonic Bloch-type mappings depend on the order of the linearly invariant family of holomorphic functions and the order of the linearly invariant family of harmonic mappings, respectively.</p>

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Linearly Invariant Families and Harmonic Bloch Mappings on Hyperbolic Domain

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • Bappaditya Bhowmik,
  • Sambhunath Sen

摘要

We extend the notions of harmonic Bloch, harmonic Bloch-type mappings, and a linearly invariant family of harmonic mappings defined in the open unit disc 𝔻 of the complex plane to a hyperbolic domain of the complex plane. Then we establish the relation between linearly invariant family of holomorphic functions with harmonic Bloch mappings and a linearly invariant family of harmonic mappings with harmonic Bloch-type mappings defined on a hyperbolic domain of the complex plane. We also prove that the bounds of the Bloch seminorms for harmonic Bloch and harmonic Bloch-type mappings depend on the order of the linearly invariant family of holomorphic functions and the order of the linearly invariant family of harmonic mappings, respectively.